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This paper pioneers the direct numerical simulation (DNS) and physical analysis in supersonic three-temperature carbon dioxide (CO2) turbulent channel flow. CO2 is a linear and symmetric triatomic molecular, with the thermal non-equilibrium three-temperature effects arising from the interactions among translational, rotational, and vibrational modes at room temperature. Thus, the rotational and vibrational modes of CO2 are addressed. The thermal non-equilibrium effect of CO2 has been modeled in an extended three-temperature kinetic model, with the calibrated translational, rotational, and vibrational relaxation time. To solve the extended kinetic equation accurately and robustly, non-equilibrium high-accuracy gas-kinetic scheme is proposed within the well-established two-stage fourth-order framework. Compared with the one-temperature supersonic turbulent channel flow, supersonic three-temperature CO2 turbulence enlarges the ensemble heat transfer of the wall by approximate 20% and slightly decreases the ensemble frictional force. The ensemble density and temperature fields are greatly affected, and there is little change in Van Driest transformation of streamwise velocity. The thermal non-equilibrium three-temperature effects of CO2 also suppress the peak of normalized root mean square of density and temperature, normalized turbulent intensities and Reynolds stress. The vibrational modes of CO2 behave quite differently with rotational and translational modes. Compared with the vibrational temperature fields, the rotational temperature fields have the higher similarity with translational temperature fields, especially in temperature amplitude. Current thermal non-equilibrium models, high-accuracy DNS and physical analysis in supersonic CO2 turbulent flow can act as the benchmark for the long-term applicability of compressible CO2 turbulence. [ABSTRACT FROM AUTHOR]

Current research conducts the quantitative comparisons between implicit large eddy simulation (iLES) and explicit eddy-viscosity large eddy simulation (eLES). iLES and eLES in a compressible Taylor–Green vortex problem are implemented with a fourth-order finite-volume gas kinetic scheme. Compared with the key statistical quantities of direct numerical simulation, iLES outweighs eLES on the exactly same unresolved grids. With DNS solution, a priori analysis of compressible filtered subgrid-scale (SGS) turbulent kinetic energy ρ ¯ K sgs f is performed. Forward and backward filtered SGS turbulent kinetic energy transfer coexists. The ensemble turbulent kinetic energy Ek is on the order of o (10 4) to o (10 2) of ensemble filtered SGS turbulent kinetic energy K sgs f . The ensemble dominant physical dissipation rate ε 1 is approximately 20 times larger than the ensemble filtered SGS dissipation rate − τ i j f S ̃ i j f . Then, for iLES and eLES, the total dissipation rate is decomposed into the resolved physical dissipation rate ε phy , modeling SGS dissipation rate ε sgs mod , and numerical SGS dissipation rate ε sgs num . Quantitative comparisons on the modeling SGS dissipation rate and numerical SGS dissipation rate in iLES and eLES are evaluated. The numerical dissipation in iLES can be treated as the built-in SGS dissipation, which accounts for the reasonable performance of iLES. While the explicit modeling SGS dissipation in eLES pollutes the resolved turbulent structures in such low-Reynolds number turbulence. The next generation of large eddy simulation on unresolved grids must take into account both the built-in numerical SGS dissipation and its competition explicit modeling SGS dissipation. [ABSTRACT FROM AUTHOR]

For simulation of hypersonic turbulent and transitional flows at high Reynolds numbers, a gas-kinetic scheme (GKS) strongly coupled with the turbulent kinetic energy equation in shear stress transport (SST) k−ω turbulence model is developed. To extend the current method to transitional flows, Langtry-Menter γ − R e θ transition model is added together with SST k−ω turbulence model. Comparisons among the numerical solutions from current GKS method coupled with turbulent kinetic energy equation, GKS method uncoupled with turbulent kinetic energy equation, Reynolds-Averaged Navier-Stokes (RANS) solutions with turbulence/transition models, and experimental data are conducted. It is concluded that the current method can significantly improve the accuracy of numerical results. [ABSTRACT FROM AUTHOR]

In recent years, coupled with traditional turbulence models, the second-order gas-kinetic scheme (GKS) has been used in the turbulent flow simulations. At the same time, high-order GKS has been developed, such as the two-stage fourth-order scheme (S2O4) GKS, and used for laminar flow calculations. In this paper, targeting on the high-Reynolds number engineering turbulent flows, an implicit high-order GKS with Lower-Upper Symmetric Gauss-Seidel (LU-SGS) technique is developed under the S2O4 framework. Based on Vreman-type LES model and k − ω SST model, a turbulent relaxation time is obtained and used for the determination of an enlarged particle collision time for the high-Reynolds number turbulent flow simulation. Numerical experiments include incompressible decaying homogeneous isotropic turbulence, incompressible high-Reynolds number flat plate turbulent flow, incompressible turbulence around NACA0012 airfoil, transonic turbulence around RAE2822 airfoil, and transonic high-Reynolds number ARA M100 wing-body simulation. Comparisons among the numerical solutions from current implicit high-order GKS, the explicit high-order GKS, the implicit second-order GKS, and experimental measurements have been conducted. Through these examples, it is concluded that the high-order GKS has high accuracy in space and time, especially for smooth flows, and obtains more accurate turbulent flow fields on coarse grids than the second-order GKS. In addition, significant acceleration on computational efficiency, as well as super robustness in simulating complex flows are confirmed from the current implicit high-order GKS. This study also indicates that turbulence modeling plays a dominant role in capturing physical solution, such as in the transonic three-dimensional complex RANS simulation, which is beyond the numerical discretization error, such as the differences between the second and fourth-order GKS. [ABSTRACT FROM AUTHOR]

For increasingly non-equilibrium flowfields, the Navier-Stokes equations lose accuracy partially due to the single temperature approximation. To overcome this barrier, a continuum multi-temperature model based on the Bhatnagar-Gross-Krook equation coupled with the Landau-Teller-Jeans relaxation model has been proposed for two-dimensional hypersonic non-equilibrium multi-temperature flow computation. In a recent study, a two-stage fourth-order gas-kinetic scheme (GKS) has been developed for equilibrium flows, which achieves a fourth-order accuracy in space and time as well as high efficiency and robustness. In this paper, targeting for accurate and efficient simulation of multi-temperature non-equilibrium flows, a high-order three-dimensional multi-temperature GKS is constructed under the two-stage fourth-order framework, with the fourth-order Simpson interpolation rule for the newly emerged source term. Simulations on decaying homogeneous isotropic turbulence, low-density nozzle flow, rarefied hypersonic flow over a flat plate, and type IV shock-shock interaction are used to validate the multi-temperature model through the comparison with experimental measurements. The unified gas kinetic scheme (UGKS) results and the Direct Simulation Monte Carlo (DSMC) solutions will be used as well in some cases for validation. Computational results not only confirm the high-order accuracy and quite robustness of this scheme but also show the significant improvement on computational efficiency compared with UGKS and DSMC, especially for the flow in the near continuum regime. [ABSTRACT FROM AUTHOR]

High-order gas-kinetic scheme (HGKS) has become a workable tool for the direct numerical simulation (DNS) of turbulence. In this paper, to accelerate the computation, HGKS is implemented with the graphical processing unit (GPU) using the compute unified device architecture (CUDA). Due to the limited available memory size, the computational scale is constrained by single GPU. For large-scale DNS of turbulence, we develop a multi-GPU HGKS simulation using message passing interface (MPI) and CUDA. The benchmark cases for compressible turbulence, including Taylor-Green vortex and turbulent channel flows, are presented to assess the numerical performance of HGKS with Nvidia TITAN RTX and Tesla V100 GPUs. For single-GPU computation, compared with the parallel central processing unit (CPU) code running on the Intel Core i7-9700 with open multi-processing (OpenMP) directives, 7x speedup is achieved by TITAN RTX and 16x speedup is achieved by Tesla V100. For multiple-GPU computation, multiple-GPU accelerated HGKS code scales properly with the increasing number of GPU. The computational time of parallel CPU code running on 1024 Intel Xeon E5-2692 cores with MPI is approximately 3 times longer than that of GPU code using 8 Tesla V100 GPUs with MPI and CUDA. Numerical results confirm the excellent performance of multiple-GPU accelerated HGKS for large-scale DNS of turbulence. Besides reducing memory access pressure, we also exploit single precision floating point arithmetic to accelerate HGKS on GPUs. Reasonably, compared to the computation with FP64 precision, the efficiency is improved and the memory cost is reduced with FP32 precision. Meanwhile, the differences in accuracy for statistical turbulent quantities appear. For turbulent channel flows, difference in long-time statistical turbulent quantities is acceptable between FP32 and FP64 precision solutions. While the obvious discrepancy in instantaneous turbulent quantities can be observed, which shows that FP32 precision is not safe for DNS in compressible turbulence. The choice of precision should be depended on the requirement of accuracy and the available computational resources. • To conduct DNS of turbulence, the HGKS is developed with single GPU+CUDA, and multiple GPUs using MPI+CUDA. • For multiple-GPU computation, multiple-GPU accelerated HGKS code scales properly with No. of GPU. • Efficiency of single Tesla V100 GPU is comparable with the MPI code using 300 CPU cores. • Efficiency of GPU code with 8 Tesla V100 GPUs approximately equals to MPI code with 3000 cores. • Efficiency is improved with FP32 precision, but differences in accuracy appear with the long-time computing. [ABSTRACT FROM AUTHOR]

The performance of high-order gas-kinetic scheme (HGKS) has been investigated for the direct numerical simulation (DNS) of isotropic compressible turbulence up to the supersonic regime [9]. Due to the multi-scale nature and coupled temporal-spatial evolution process, HGKS provides a valid tool for the numerical simulation of compressible turbulent flow. Based on the domain decomposition and message passing interface (MPI), a parallel HGKS code is developed for large-scale computation in this paper. The standard tests from the nearly incompressible flow to the supersonic one, including Taylor-Green vortex problem, turbulent channel flow and isotropic compressible turbulence, are presented to validate the parallel scalability, efficiency, accuracy and robustness of parallel implementation. The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme, including the resolution of flow structure and efficiency of computation. Based on the accuracy of the numerical solution, the numerical dissipation of the scheme in the turbulence simulation is quantitatively evaluated. Meanwhile, based on the kinetic formulation HGKS shows advantage for supersonic turbulent flow simulation with its accuracy and robustness. The current work demonstrates the capability of HGKS as a powerful DNS tool from the low speed to supersonic turbulence study, which is less reported under the framework of finite volume scheme. • A parallel HGKS code is developed as a powerful DNS tool from the low speed to supersonic turbulence study. • The parallel scalability, efficiency, accuracy and robustness of parallel implementation of HGKS are validated. • The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme. • As a mesoscopic method, HGKS also performs better than both lattice Boltzmann method (LBM) and discrete unified gas-kinetic scheme (DUGKS). [ABSTRACT FROM AUTHOR]

• The DNS solution on a turbulent Mach number up to 2 has been obtained by high-order gas-kinetic scheme. • SGS production, dissipation, dilation, and diffusion have been evaluated. • The coarse-graining analysis gives the order of magnitude of all SGS terms in compressible K s g s budget. The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime with M a t = 1.2 has been investigated by the high-order gas-kinetic scheme (HGKS) (Cao et al. (2019) [8]). In this study, the DNS on a much higher initial turbulent Mach number up to M a t = 2.0 is performed by HGKS, which confirms the super robustness of HGKS. The coarse-graining analysis of subgrid-scale (SGS) turbulent kinetic energy K s g s budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The exact compressible SGS turbulent kinetic energy K s g s transport equation is derived with density weighted filtering process. Based on the compressible K s g s transport equation, the coarse-graining processes are implemented on three sets of unresolved grids with the Box filter. The coarse-graining analysis of compressible K s g s budgets shows that all unresolved source terms are dominant in the current system. Especially, the magnitude of SGS pressure-dilation term is on the order of SGS solenoidal dissipation term within the initial acoustic time scale. Therefore, the SGS pressure-dilation term cannot be neglected as that in the previous work. The delicate coarse-graining analysis of SGS diffusion terms in compressible K s g s equation also confirms that both the fluctuation velocity triple correlation term and the pressure-velocity correlation term are dominant terms. The current analysis provides an indication on the order of magnitude of all SGS terms in compressible K s g s budget, which provides a solid basis for compressible LES modeling of high Mach number turbulent flow. [ABSTRACT FROM AUTHOR]

In this study, the high-order gas kinetic scheme (GKS) is employed for explicit large eddy simulation (hereafter referred to simply as "LES") and implicit large eddy simulation (iLES) of turbulent channel flows. The main objective is to compare the performance of iLES and LES in the high-order finite volume framework, and study which is most suitable for turbulence simulation. The prediction abilities of different explicit LES models and iLES method on various statistics and flow structures are compared. Most results from both iLES and explicit LES are very close to those of direct numerical simulation. Moreover, iLES is generally superior to explicit LES in predicting several important flow properties, including the mean velocity profiles, Reynolds stress, and Q-criterion iso-surfaces. This superior performance of iLES may arise that the numerical dissipation of the high-order scheme is enough to replace the subgrid dissipation needed in large eddy simulation of turbulence. If the explicit LES model is further adopted, the dissipation will increase, so the results degrade. The overall satisfactory results show that the high-order GKS can provide appropriate numerical dissipation and is suitable for iLES of turbulence. [ABSTRACT FROM AUTHOR]